Fitting a modified gaussian

Hello Experts,

I am new to data fitting. I would be very gratefull for a detailed response. I have a data set that looks like a streched gaussian distribution, or a reverse log normal distribution please see attached plot. Please let me know what distribution will be the best to get a good fit for this type of data. If you could please provide me with an example that would be very helpful. I have attached x and y data for your reference.

Thanks very much!

load x
load y
plot(x, y)

Answer 1

[Mathieu NOE]

hello

I am by no mean a curve fitting or stats expert but this is what I could achieve :

this is a reverse log normal distribution - so basically a gaussian fit realized not on x but on exp(x/constant)

f = @(a,b,c,d,x) a.*exp(-(exp(x/d)-b).^2 / c.^2);

plot

code

load('x.mat');
load('y.mat');

% curve fit using fminsearch
f = @(a,b,c,d,x) a.*exp(-(exp(x/d)-b).^2 / c.^2);
obj_fun = @(params) norm(f(params(1), params(2), params(3), params(4),x)-y);

d_init = 1000; 
[a_init,ind] = max(y);
b_init = exp(x(ind)/d_init);

sol = fminsearch(obj_fun, [a_init,b_init,b_init/2,d_init]);
a_sol = sol(1)
b_sol = sol(2)
c_sol = sol(3)
d_sol = sol(4)
xx = linspace(min(x),max(x),200);
y_fit = f(a_sol, b_sol,c_sol,d_sol, xx);
yy = interp1(x,y, xx);
Rsquared = my_Rsquared_coeff(yy,y_fit); % correlation coefficient

plot(xx, y_fit, '-',x,y, 'r .', 'MarkerSize', 15)

title(['Gaussian Fit / R² = ' num2str(Rsquared) ], 'FontSize', 15)
ylabel('Amplitude', 'FontSize', 14)
xlabel('x', 'FontSize', 14)
eqn = " y = "+a_sol+ " * exp(-(exp(x / " +d_sol+"  )- " +b_sol+")² / (" +c_sol+ ")²";
legend(eqn)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function Rsquared = my_Rsquared_coeff(data,data_fit)
    % R2 correlation coefficient computation
    
    % The total sum of squares
    sum_of_squares = sum((data-mean(data)).^2);
    
    % The sum of squares of residuals, also called the residual sum of squares:
    sum_of_squares_of_residuals = sum((data-data_fit).^2);
    
    % definition of the coefficient of correlation is
    Rsquared = 1 - sum_of_squares_of_residuals/sum_of_squares;
    
    
end

[Sanchit Sharma]

Hello @Mathieu NOE,

Thanks very much! This was very helpful. Could you please elaborate the meaning of line:

obj_fun = @(params) norm(f(params(1), params(2), params(3), params(4),x)-y);

Also, why did you assume c_init = b_init/2?

Best,

S

[Mathieu NOE]

hello again

for the first question, I would simply recommend to see the fminsearch documentation. Simply follow how the function to minimize is defined (handle);

Find minimum of unconstrained multivariable function using derivative-free method - MATLAB fminsearch - MathWorks France

c_init = b_init/2 : a rough estimate based on how narrow or wide is the experimental peak. You can probably try other ratios , the optimizer will still converge to the optimal value

All the best !

M